The Correct Derivation of the Buckling Equations of the Shear-Deformable FGM Plates for the Extended Kantorovich Method

Abstract

This article presents the derivation of the elastic buckling equations and boundary conditions of shear-deformable plates in the frame of the extended Kantorovich method (EKM). Surveying the literature shows that those stability equations are often obtained using a wrong derivation by confusing them with the linear equilibrium condition. This work aims at providing the correct derivation that is built on the stability of the equilibrium condition. Buckling equations are derived for three different plate theories, namely, the first-order shear deformation plate theory (FSDT), the refined-FSDT, and the refined plate theory (RPT). This article is the first to implement the EKM based on a refined theory. Also, it is the first time to implement the refined-FSDT in buckling analysis. For the generic FGM plates, buckling equations derived based on the FSDT and refined-FSDT are both found to be simple and contain only the lateral displacements/rotations variations. On the other hand, those of the RPT, have coupled lateral and in-plane displacement variations, even if the physical neutral plate is taken as the reference plane. The considered plate is rectangular and under general in-plane loads. The properties of the plate are continuously varying through its thickness which is assumed to change smoothly with a separable function in the two in-plane directions. The von Karman nonlinearity is considered. The stability equations are derived according to the Trefftz criterion, using the variational calculus. The solution methods of the obtained equations are out of the scope of this article, however, a brief on the solution strategy is presented.